Haag duality for Kitaev's quantum double model for abelian groups

TL;DR

This paper proves Haag duality for the quantum double model with abelian groups, enhancing the understanding of locality and superselection sectors in algebraic quantum field theory.

Contribution

It establishes Haag duality for conelike regions in Kitaev's quantum double model for finite abelian groups, strengthening the locality property.

Findings

Haag duality holds for the ground state representation.

Superselection sectors correspond to irreducible representations of the quantum double.

The result parallels the structure found in the toric code model.

Abstract

We prove Haag duality for conelike regions in the ground state representation corresponding to the translational invariant ground state of Kitaev's quantum double model for finite abelian groups. This property says that if an observable commutes with all observables localised outside the cone region, it actually is an element of the von Neumann algebra generated by the local observables inside the cone. This strengthens locality, which says that observables localised in disjoint regions commute. As an application we consider the superselection structure of the quantum double model for abelian groups on an infinite lattice in the spirit of the Doplicher-Haag-Roberts program in algebraic quantum field theory. We find that, as is the case for the toric code model on an infinite lattice, the superselection structure is given by the category of irreducible representations of the quantum double.